INTRODUCTION TO ENDOSCOPY
Snowbird Lectures
Utah, June 2006
Jean-Pierre Labesse
Institut Mathématique de Luminy, UMR 6206
Marseille, France
1
Endoscopy
Foreword
Many questions about non commutative Lie groups boil down to questions in invariant
harmonic analysis, the study of distributions on the group that are invariant by conjugacy.
The fundamental objects of invariant harmonic analysis are orbital integrals and characters,
respectively the geometric and spectral sides of the trace formula.
In the Langlands program a cruder form of conjugacy called stable conjugacy plays a
role. The study of Langlands functoriality often leads to correspondences that are defined
only up to stable conjugacy. Endoscopy is the name given to a series of techniques aimed
to investigate the difference between ordinary and stable conjugacy.
Let G be a reductive group over a field F . Recall that one says that γ and γ 0 in G(F )
are conjugate if there exists x ∈ G(F ) such that γ 0 = xγx−1 . Roughly speaking, stable
conjugacy amounts to conjugacy over the algebraic closure F : at least for strongly regular
semisimple elements, one says that γ and γ 0 in G(F ) are stably conjugate if there is
x ∈ G(F ) such that γ 0 = xγx−1 .
On the geometric side, the basic objects of stably invariant harmonic analysis are stable
orbital integrals. On the spectral side, the notion of L-packets of representations is the
stable analogue for characters of tempered representations. The case of non tempered
representations is the subject of conjectures of Arthur that will be examined in Vogan’s
lectures.
The word “endoscopy” has been coined to express that we want to see ordinary conjugacy
inside stable conjugacy. We shall introduce the basic notions of local endoscopy: κorbital integrals, endoscopic groups, endoscopic transfer of orbital integrals and its dual
for characters with an emphasis on the case of real groups, following the work of Diana
Shelstad.
2
Endoscopy
I.
GL(2, R)
versus
3
SL(2, R)
Endoscopy – I.1
I.1 – Representations of GL(2, R)
By representation we understand admissible representations. Their classification is as
follows: all admissible irreducible representations of GL(2, R) are subquotients of principal
series
ρ(µ1 , µ2 )
where µi are characters of R× . They are induced from the Borel subgroup i.e. it is the
right regular representation in the space of smooth functions such that
α
f(
0
x
β
1/2
α
g) = µ1 (α)µ2 (β) f (g)
β
We have three types of subquotients according to the value of
µ = µ1 µ−1
2
We shall restrict ourselves to representations that induce unitary characters on the center
and hence we assume that the product µ1 µ2 is unitary.
1 – Irreducible principal series
π(µ1 , µ2 )
where
µ 6= xn . sign (x)
for some n ∈ Z − {0}. These representations are unitarizable if µ is unitary or if µ = |x|s
with s real and −1 < s < 1.
2 – Finite dimensional subquotients
π(µ1 , µ2 )
when µ = xn . sign (x) with n ∈ Z − {0}. It is unitarizable if n = ±1.
3 – Discrete series subquotients
σ(µ1 , µ2 )
when µ = xn . sign (x) with n ∈ Z − {0}. These representations are unitarizable.
The various representations are equivalent under permutation of the µi : π(µ1 , µ2 ) '
π(µ1 , µ2 ) etc...
4
Endoscopy – I.2
I.2 – Langlands parameters for GL(2, R)
We first have to introduce the Weil group for R. This is the subgroup matrices in SU (2)
generated by
z 0
0 −1
×
z=
, z∈C
and wσ =
0 z
1 0
The element wσ acts by conjugacy as the non trivial element in the Galois group, in other
words there is an exact sequence
1 → C× → WR → Gal(C/R) → 1
and a section of the map WR → Gal(C/R) is defined by
σ 7→ wσ
Observe that wσ2 = −1 and hence the above extension of C× by Gal(C/R) is the non trivial
one.
A Langlands parameter for GL(2, R) is a conjugacy classes of homomorphisms of WR in
GL(2, C) with semisimple images.
For z = ρeiθ let χs,n (z) = ρs einθ then, up to conjugacy, the admissible maps ϕ are of the
following form:
1 – for some si ∈ C and mi ∈ Z2
ϕs1 ,m1 ,s2 ,m2 (z) =
with
χs1 ,0 (z)
0
0
χs2 ,0 (z)
ϕs1 ,m1 ,s2 ,m2 (wσ ) =
(−1)m1
0
0
(−1)m2
Up to conjugacy ϕs1 ,m1 ,s2 ,m2 ' ϕs2 ,m2 ,s1 ,m1 .
2 – for some s ∈ C and n ∈ Z
ϕs,n (z) =
χs,n (z)
0
0
χs,−n (z)
with
ϕs,n (wσ ) =
0
1
(−1)n
0
Up to conjugacy ϕs,n ' ϕs,−n .
The intersection of the two sets of conjugacy classes of maps is the class of parameters
of the form
ϕs,0 ' ϕs,1,s,0 ' ϕs,0,s,1
5
Endoscopy – I.2
Let us denote by ε the homomorphism from WR to C× defined by
and ε(wσ ) = −1
ε(z) = 1
Lemma I.1 – If ϕ is a Langlands parameter, then
ϕ⊗ε'ϕ
if and only if ϕ is in the class of ϕs,n for some s and some n.
Proof: First it is clear that
ϕs,n ⊗ ε = α ϕs,n α
−1
where
α=
−1
0
0
1
Conversely we have to observe that
ϕs1 ,m1 ,s2 ,m2 6' ϕs1 ,m1 +1,s2 ,m2 +1
unless s1 = s2 and m1 + m2 ≡ 1
The correspondence between irreducible representations and Langlands parameters for
GL(2, R) is obtained as follows. Recall the correspondence for GL(1, R) (also called Tate
Nakayama duality): a character µ of GL(1, R) = R× of the form
µ(x) = |x|s sign (x)m
correspond to a character of the Weil group
z 7→ χs,0
and wσ 7→ (−1)m
We get a natural bijection between equivalence classes of admissible irreducible representations of GL(2, R) and conjugacy classes of admissible homomorphisms of WR in GL(2, C)
as follows:
π(µ1 , µ2 ) 7→ ϕs1 ,m1 ,s2 ,m2
with
µi (x) = |x|si sign (x)mi
and
σ(µ1 , µ2 ) 7→ ϕs,n
with
µ1 µ2 (x) = |x|2s sign (x)n+1
and
n
µ1 µ−1
2 (x) = x sign (x)
A parameter corresponds to a tempered representation if the image of the map is bounded
i.e. if the si are imaginary.
6
Endoscopy – I.3
I.3 – Representations of SL(2R)
The representations of SL(2, R) and their L-packets are easily understood in terms of those
for GL(2, R). In fact it is an easy exercise to show that any irreducible representation
of SL(2, R) occurs in the restriction of an irreducible representation of GL(2, R). Those
restrictions either remain irreducible (which is the case for principal series for generic values
of the parameter) or split into two irreducible components whose union is an L-packet for
SL(2, R).
Two representations π and π 0 are in the same L-packet if and only if, up to equivalence,
they are conjugated by α:
0
π ' π ◦ Ad(α)
where
α=
−1
0
0
1
We have the following classification:
1 – Irreducible principal series: they are the π(µ) obtained by restriction of π(µ1 , µ2 )
with µ 6= xn . sign (x) with n ∈ Z.
2 – Finite dimensional representations of dimension n: they are the π(µ) obtained by
restriction of π(µ1 , µ2 ) with µ = xn . sign (x) and n 6= 0.
3 – Discrete series L-packets
+
−
σ(µ) = {D|n|
, D|n|
}
are obtained by restriction of σ(µ1 , µ2 ) with
µ = xn . sign (x)
and n 6= 0.
4 – Limits of discrete series L-packet
σ(µ) = {D0+ , D0− }
is obtained by restriction of π(µ1 , µ2 ) with
µ = sign (x)
The L-packets of representations that are indexed by characters µ and µ−1 are equivalent.
The minimal K-type for Dn± is ±(n + 1) i.e.
r(θ) 7→ exp (±i (n + 1) θ)
7
Endoscopy – I.3
if
r(θ) =
cos θ
− sin θ
sin θ
cos θ
The character of Dn+ on K = SO(2, R) is given by
Θ+
n (r(θ)) =
ei(n+1)θ
−einθ
=
1 − e2iθ
eiθ − e−iθ
while the character of Dn− is the complex conjugate:
Θ−
n (r(θ)) =
e−i(n+1)θ
e−inθ
=
1 − e−2iθ
eiθ − e−iθ
8
Endoscopy – I.4
I.4 – Langlands parameters for SL(2, R)
From the bijection between equivalence classes of representations and conjugacy classes
of Langlands parameters for GL(2, R) one derives a bijection between equivalence classes
of L-packets of admissible irreducible representations of SL(2, R) and conjugacy classes of
admissible homomorphisms of WR in P GL(2, C). We need the
Lemma I.2 – Any projective representation of WR lifts to a representation.
Proof: This is a particular case of a result of [Lab1]. This particular case can also be
found in [Lan2]. We shall prove it only for two dimensional representations. Consider a
two-dimensional projective parameter. The image of C× is inside a torus. Now this image
is either trivial or is a maximal torus in P GL(2, C) and hence of dimension 1. If this image
is trivial the lemma is easy to prove. If this image is a torus the map restricted to C× is
given by a non trivial character χs,n . If n = 0 th existence of a lift is easy. If n 6= 0 then
and the image of wσ must lie in the normalizer of the torus and acts non trivially and
hence its square is central i.e. projectively trivial and s = 0. Since wσ2 = −1 this shows
that n is even and hence ϕ0,n/2 is a lift.
The correspondence is a follows:
1 – the parameter for π(µ) is the conjugacy class of the projective parameter ϕs,m defined
by ϕs,m,0,0 with
µ(x) = |x|s sign (x)m
2 – the parameter for Dn± is the conjugacy class of the projective parameter ϕn defined
by ϕ0,n .
We have seen in lemma I.1 that
ϕ0,n ⊗ ε = αϕ0,n α
−1
where
α=
−1
0
0
1
But ε has a central image and hence the projective parameters defined by ϕ0,n and ϕ0,n ⊗ε
are equal. This shows that the projective image of α belongs to the centralizer of the
projective image of ϕ0,n .
Let ϕn be the projective parameter defined by ϕ0,n and let Sϕn denote the centralizer of
the image of ϕn and Sϕn the quotient of Sϕn by its connected component Sϕ0 n times the
center ZǦ of Ǧ (trivial here). When n 6= 0 we have
Sϕn = Sϕn ' {1, α}
When n = 0 the group Sϕ0 0 is a torus but again Sϕ0 is generated by the image of α.
9
Endoscopy
II. Endoscopy: the simplest case
10
Endoscopy – II.1
II.1 – Endoscopy for SL(2, R)
We shall first discuss endoscopy for SL(2). This is the simplest instance of endoscopy
and in fact it is in this case that endoscopy was first observed. As far as I know (but
Bill Casselman may know more), endoscopy was discovered by Langlands while he was
trying to understand the Zeta function of some simple Shimura varieties: those attached
to compact inner forms of groups between GL(2) and SL(2) over a totally real field.
Endoscopy arises because the conjugacy over F and over F (the algebraic closure of F )
may be different. For example inside SL(2, R) the two rotations
r(θ) =
and
cos θ
− sin θ
r(−θ) =
sin θ
cos θ
− sin θ
cos θ
cos θ
sin θ
are not conjugate (unless θ ∈ Z π) although they are conjugate inside SL(2, C) and also
inside GL(2, R) by
−i 0
−1 0
w=
and α =
0 i
0 1
The union of the sets of conjugates of elements in the pair (r(θ), r(−θ)) is said to be a
stable conjugacy class. Similarly the unipotent elements
u0 =
1
0
1
1
and
u−1
0
=
1 −1
0
1
are conjugate in SL(2, C) but not in SL(2, R). There are two conjugacy classes and only
one stable conjugacy class of (non trivial) unipotent elements.
Endoscopy on the spectral side is also easily observed for SL(2, R): discrete series and
limits of discrete series representations come by pairs called L-packets.
In all above examples the objects inside pairs are exchanged by conjugacy under w = iα
an element in the normalizer of SO(2) in SL(2, C). We observe that if σ is the non trivial
element of the Galois group
−1
aσ = wσ(w)
=
−1
0
0 −1
generates a subgroup of order 2 that can be identified with H 1 (C/R, SO(2)). The characters of this 2-group are called endoscopic character. The endoscopic groups for SL(2, R)
correspond to these two characters. The “trivial” endoscopic group is SL(2, R) himself
while the non trivial one is T (R) = SO(2, R) the compact torus.
11
Endoscopy – II.1
Finally we observe that the sum
SΘn (r(θ)) =
Θ+
n (r(θ))
+
Θ−
n (r(θ))
einθ − e−inθ
= − iθ
e − e−iθ
is invariant under conjugacy by w, and we say that SΘn is a stable character, while
−
inθ
∆(r(θ))(Θ+
+ e−inθ
n − Θn )(r(θ)) = e
is the sum of two characters of T (R) where
∆(r(θ)) = −2i sin θ
For further generalization we observe that
∆(r(θ)) = −eiθ (1 − e−2iθ ) = −i sign (sin(θ))|eiθ − e−iθ |
The group G∗ = SL(2) over R has an inner form G such that G(R) = SU (2). Observe
that SL(2, R) and SU (2) have in common the compact Cartan subgroup
T (R) = SO(2)
The conjugacy classes in SU (2) are classified by the pairs of eigenvalues {eiθ , e−iθ }. Hence
they are in bijection with elliptic stable conjugacy classes in SL(2, R). This correspondence
is dual to the correspondence between representation Fn of dimension n of SU (2) and Lpackets of discrete series Dn± and there is the character identity
trace Fn (r(θ)) = −SΘn (r(θ))
One should observe that inside GL(n, F ) conjugacy over F and over F coincide. Hence
there is no non trivial endoscopy for GL(n) in the sense that conjugacy and stable conjugacy coincide and L-packets are singletons. Nevertheless, if we consider now an inner
forms G of G∗ = GL(n), then there is a correspondence for conjugacy classes and for representations between G and G∗ . These correspondences, often called Jacquet-Langlands
correspondences, are nowadays considered as an instance of endoscopy.
12
Endoscopy – II.2
II.2 – Asymptotic behaviour of orbital integrals
Let f be a smooth and compactly supported function on G = SL(2, R). We are to study
its orbital integrals. For γ ∈ G let us denote by Gγ its centralizer in G. The orbital integral
is
Z
Oγ (f ) =
f (x−1 γx) dẋ
Gγ \G
Observe it depends on the choice of Haar measures on G and Gγ . We shall study the
asymptotic behaviour of Oγ (f ) when γ is diagonal and γ → 1 and when γ = r(θ) and
θ → 0. We may and will assume that f is K-central i.e. f (kx) = f (xk).
Consider the first case where γ is diagonal:
a 0
γ=
0 b
with ab = 1. Then
Z
f (u−1 γu) du
Oγ (f ) =
U
for a standard choice of Haar measures, where U is the group of unipotent matrices and
hence
Z a (a − b)x
Oγ (f ) =
f
dx
0
b
R
which yields the
Lemma II.1 – Let ∆(γ) = |a − b| then
h(γ) = ∆(γ)Oγ (f )
extends to a smooth function on the group A(R) of diagonal real matrices.
This is the simplest example of endoscopic geometric transfer for a non elliptic endoscopic
group.
When γ = r(θ) the orbital integral can be computed using the Cartan decomposition
G = KAK. Here we may take K = T (R) the group of rotations. We have
Or(θ) (f ) = c F (sin θ)
with c a constant that depends on the choice of Haar measures and
Z ∞ a(λ)
tλ
t − t−1 dt
F (λ) =
f
−1
−t λ a(λ)
t
1
where
a(λ) =
p
1 − λ2
13
Endoscopy – II.2
Observe that since f is K-central we have for any a, b and c
a b
a −c
f
=f
c a
−b a
and hence
∞
Z
ε(t − 1)f
F (λ) =
0
a(λ)
−t−1 λ
tλ
a(λ)
dt
with ε(x) = sign(x). To study the asymptotic its behaviour when λ → 0 we consider
Z ∞
a(λ) λt
A(λ) =
ε(t − 1)f
dt
0
a(λ)
0
By Taylor-Lagrange formula
F (λ) = A(λ) + λB(λ)
where
∞
Z
ε(t − 1)g
B(λ) =
0
a(λ)
−t−1 λ
tλ
a(λ)
dt
t
for some smooth function g compactly supported in the upper right variable and that has
a O(u)−1 decay in the lower left variable u so that the integral is absolutely convergent.
Observe that
−1
Z
∞
A(λ) = |λ|
f
0
1 ε(λ)u
0
1
du − 2f (1) + o(λ)
Since B(λ) is at most of logarithmic growth we see that the even functions
G(λ) = |λ|(F (λ) + F (−λ))
and
H(λ) = λ(F (λ) − F (−λ))
extend to continuous functions at the λ = 0.
We need more precise informations on the asymptotic behaviour. Hence we have to look
more carefully to the error term B(λ) which, we recall, equals:
Z
∞
ε(t − 1)g
0
a(λ)
−t−1 λ
tλ
a(λ)
dt
t
But this is the difference of two terms whose leading terms are equivalent to ln(|λ|−1 )g(1)
up to continuous terms. Hence B is continuous. Generalizing this process one gets asymptotic expansions of the following form:
G(λ) =
N
X
an |λ|−1 + bn λ2n + o(λ2N )
n=0
14
Endoscopy – II.2
and
N
X
H(λ) =
hn λ2n + o(λ2N )
n=0
and hence H(λ) is smooth. We have proved the following
Lemma II.2 – There is a smooth fonction h on T (R) such that:
h(γ) = ∆(γ) Oγ (f ) − Ow(γ) (f )
for γ = r(θ) ∈ T (R) and
∆(r(θ)) = −2i sin θ
This lemma establishes the simplest case of a non trivial geometric endoscopic transfer
for an elliptic endoscopic group. A variant of this lemma establishes the transfer between
orbital integrals weighted by the character of order two
ω = sign ◦ det
for functions on GL(2, R):
Oγω (f )
Z
ω(x)f (x−1 γx) dẋ
=
Gγ \G
and functions on the elliptic maximal torus
a
×
Te(R) ' C =
−b
b
a
∈ GL(2, R)
Lemma II.3 – There is a smooth fonction h on Te(R) such that:
h(γ) = ∆(γ)Oγω (f )
for γ = ρ r(θ) ∈ Te(R) and
∆(r(θ)) = −2i sin θ
An other simple example is given by the transfer between inner forms.
Consider the compact group G = SU (2, R). This is an inner form of G∗ = SL(2, R). In
fact one has g ∈ SU (2, R) iff
g ∈ SL(2, C)
and g = Jσ(g)J −1
15
Endoscopy – II.2
with
J=
0 −1
1
0
and σ(g) is the complex conjugate of g.
There is an injection of the set of conjugacy classes inside G into the set of stable conjugacy classes in G∗ induced by the following correspondence: given γ ∈ SU (2, R) we denote
by γ̃ any semisimple element γ̃ ∈ SL(2, R) with the same eigenvalues.
Lemma II.4 – Given f ∈ Cc∞ (SU (2, R)) there is a function f˜ ∈ Cc∞ (SL(2, R)) such that
for γ ∈ SU (2, R) non central
SOγ̃ (f˜) = −Oγ (f )
where
SOγ (f˜) =
X
SOw(γ̃) (f˜)
and the sum is over the complex Weyl group for SO(2) ⊂ SL(2) and
SOγ̃ (f˜) = Oγ̃ (f˜) = 0
if γ̃ ∈ SL(2, R) has distinct real eigenvalues.
An easy proof is obtained using pseudo-coefficients. This will be explained in a fairly
general case in a section toward the end of these notes.
In the case of GL(2) a similar result is used for all local fields in Jacquet-Langlands. This
lemma is a particular case of the general result due to Shelstad [She1] one the transfer
between inner forms of real groups.
16
Endoscopy
III. The general case
17
Endoscopy – III.1
III.1 – Group cohomology and hypercohomology
Consider first a set X with an action of a group Γ. By definition H 0 (Γ, X) is the fixed
point set X Γ : the set of x ∈ X such that
x = σ(x)
for all σ ∈ Γ
Now let G be an algebraic group with an action of Γ compatible with the group structure.
Then H 0 (Γ, G) is a group. Moreover we may define a set H 1 (Γ, G) as the quotient of the
set Z 1 (Γ, G) of 1-cocycles i.e. maps
σ ∈ Γ 7→ aσ ∈ G
such that
aσ σ(aτ )a−1
στ = 1
by the equivalence relation
a0σ ' aσ ⇐⇒ a0σ = b aσ σ(b)−1
for some b ∈ G. The set H 1 (Γ, G) has no natural group structure. But the trivial class
makes it a pointed set. Finally, if A is an abelian group, we have the familiar cohomology
groups in all degrees
H i (Γ, A)
This can be slightly generalized as follows. One can define a hyper-cohomological theory
in degrees ≤ 0 for complexes of the form
[G ⇒ X]
where G is a group acting (on the left) on a set X with a group Γ acting on the complex.
The cohomology set
H 0 (Γ, G ⇒ X)
it the quotient of the set of pairs (x, a) with x ∈ X and a a map from Γ to X with
x = aσ . σ(x)
and aσ σ(aτ )a−1
στ = 1
by the equivalence relation
(x0 , a0 ) ' (x, a) ⇐⇒ x0 = bx and a0σ = b aσ σ(b)−1
There is also an hyper-cohomological theory in degrees ≤ 1 for complexes of groups of
the form
[G0 → G]
18
Endoscopy – III.1
that are “left crossed modules”. This means that G and G0 are groups and we are given
not only a homomorphism G0 → G, which yields an action of G0 on G by left translations,
but also an action of G on G0 and that those two actions are compatible with adjoint
actions. This implies in particular that
ker[G0 → G]
is an abelian group.
A morphism of crossed modules
[G0 → G] → [H 0 → H]
is a pair of maps G0 → H 0 and G → H that intertwines the various actions, in particular
we have a commutative diagram
G0 → G
↓
↓
0
H → H
Such a morphism is a quasi-isomorphism if it induces isomorphisms for kernels and cokernels.
If we have an action of a group Γ, compatible with the structure of crossed module, one
has a cohomology theory
H i (Γ, G0 → G)
in degrees i ≤ 1. The H 1 is defined as a quotient of the set of 1-hyper-cocycles (a, b) with
aσ σ(aτ )a−1
στ = ρ(bσ,τ )
where ρ is the homomorphism G0 → G and
aσ ∗ σ(bτ,µ ) . bσ,τ µ = bσ,τ . bστ,µ
where ∗ denotes the action of G on G0 . The equivalence relation is
a0σ = α ρ(βσ )aσ σ(α)−1
and
−1
b0σ,τ = α ∗ (βσ (aσ ∗ σ(βτ ))bσ,τ βστ
)
There is a natural group structure on H 0 but again H 1 is simply a pointed set while
H −1 (Γ, G0 → G) := H 0 (Γ, ker[G0 → G])
19
Endoscopy – III.1
is an abelian group. These cohomological objects are functorial for the category of crossed
modules and, as for usual hypercohomology theories, they are invariant under quasiisomorphisms. Cohomology in higher degrees cannot be defined (in a functorial way)
without further structures.
We observe that G can be seen as the crossed module
[1 → G]
and we have
H i (Γ, G) = H i (Γ, 1 → G)
There is an exact sequence
1 → H −1 (Γ, G0 → G) → H 0 (Γ, G0 ) → H 0 (Γ, G) →
H 0 (Γ, G0 → G) → H 1 (Γ, G0 ) → H 1 (Γ, G) → H 1 (Γ, G0 → G)
with
H −1 (Γ, G0 → G) := H 0 (Γ, ker ρ)
The exactness has to be understood for pointed sets. In general the exact sequence cannot
be continued since the H 2 does not make sense for a non commutative group.
If we denote by K the kernel of [G0 → G] and by L the cokernel we have another exact
sequence:
1 → H 0 (Γ, K) → H −1 (Γ, G0 → G) → H −1 (Γ, L) →
H 1 (Γ, K) → H 0 (Γ, G0 → G) → H 0 (Γ, L) → H 2 (Γ, K)
→ H 1 (Γ, G0 → G) → H 1 (Γ, L) → H 3 (Γ, K)
where H −1 (Γ, H) = 1. Observe that since K is abelian the H i (Γ, K) are defined for all i.
If moreover the complex G0 → G is quasi-isomorphic to a complex of abelian groups
B → A (in a way compatible with Γ-actions) then one can define hyper-cohomology groups
H i (Γ, G0 → G) := H i (Γ, B → A)
in all degrees and the above exact sequence extends indefinitely.
20
Endoscopy – III.2
III.2 – Galois cohomology and abelianized cohomology
Let F be a field and let F be an separable closure. Consider an algebraic variety X defined
over F then the set X(F ) of points with value in F is the fixed point set of Gal(F /F ) on
X(F ). This is the 0-th cohomology set of the Galois group with values in X(F ):
X(F ) = H 0 (Gal(F /F ), X(F ))
We shall also use the standard notation for Galois cohomology
H ∗ (F, X) := H ∗ (Gal(F /F ), X(F ))
whenever defined. If G is an algebraic group over F we have also the set
H 1 (F, X) := H 1 (Gal(F /F ), X(F ))
Finally, if A is an abelian algebraic group we have
H i (F, X) := H i (Gal(F /F ), X(F ))
for all i.
Let G be a connected reductive group over a field F and let GSC be the simply connected
cover of its derived subgoup. We observe that complexes
[GSC → G]
are crossed modules for the obvious adjoint actions and hence we have at hand
H ∗ (F, GSC → G)
in degrees ≤ 1. But in fact such a complex is quasi-isomorphic to a complex of abelian
groups:
[Zsc → Z]
where Z is the center of G and Zsc is the center of GSC . Since hypercohomology is invariant
by quasi-isomorphisms this allows to define abelian groups
H ∗ (F, GSC → G) = H ∗ (F, Zsc → Z)
in all degrees. Another quasi-isomophic complex of abelian groups is useful: let T be a
torus in G and let Tsc its preimage in GSC then
H ∗ (F, GSC → G) = H ∗ (F, Tsc → T )
21
Endoscopy – III.2
We shall use the following compact notation
∗
Hab
(F, G) := H ∗ (F, GSC → G)
Theorem III.1 – Let G be a connected reductive group over a field F and let GSC be
the simply connected cover of its derived subgoup. There is a family of abelian group
∗
Hab
(F, G) with natural maps
i
H i (F, G) → Hab
(F, G) for i ≤ 1
giving rise to a long exact sequence
0
1
→ Hab
(F, G) → H 1 (F, GSC ) → H 1 (F, G) → Hab
(F, G)
Moreover, when F is a local field
1
H 1 (F, G) → Hab
(F, G)
is surjective and even bijective if F is non-archimedean.
Proof: . The surjectivity follows from the existence of fundamental tori and that such tori
have a vanishing H 2 . The injectivity follows from a theorem due to Kneser quoted below.
Theorem III.2 – When F is a non-archimedean local field then
H 1 (F, GSC ) = 1
We observe that using the universal property of simply connected spaces one can show
that the abelianization map
1
H 1 (F, G) → Hab
(F, G)
is functorial in G (cf.[Lab3] lemme I.6.3)
Before the introduction of abelianized cohomology by Borovoi [Bo] (see also [Lab3]),
Kottwitz had found a substitute for the abelianization map but that was more subtle to
define, not obviously functorial and was restricted to reductive groups over local fields.
Given G reductive over some field F consider the abelian group
π0 (Z(Ǧ)Γ )
where Ǧ is the complex dual group, Z(Ǧ) its center, Γ the Galois group Gal(F /F ) and
π0 the group of connected components. Now, using Tate-Nakayama duality, Kottwitz has
shown that when F is a local field there exists a canonical map
H 1 (F, G) → π0 (Z(Ǧ)Γ )D
22
Endoscopy – III.2
where the exponent D denotes the Pontryagin duality. Using abelianized cohomology one
recovers Kottwitz’s map using that
[Ǧ → (GSC )ˇ]
is quasi-isomorphic to
[Z(Ǧ) → 1]
and by Tate-Nakayama duality one gets an injective homomorphism
1
Hab
(F, G) → π0 (Z(Ǧ)Γ )D
which is bijective when F is non archimedean ([Lab3] Proposition 1.7.3).
We still have to introduce the Kottwitz signs [Ko1]. Given a reductive group G consider
its quasisplit inner form G∗ and G∗ad the adjoint group. Let a(G) be the cohomology class
in H 1 (F, G∗ad ) defining G as an inner form of G∗ . There is an isomorphism from
∗
Hab
(F, G∗ad ) → H 2 (F, Z(G∗SC ))
and by composition we get a class
a(G) ∈ H 2 (F, Z(G∗SC ))
Now the half sum of positive roots (for some order) defines a map from Z(G∗SC ) to the
group of elements of order 2 in the multiplicative group and hence if F is local we get a
number e(G) ∈ {±1} called the Kottwitz sign. It can be shown that when F = R one has
e(G) = (−1)q(G
∗
)−q(G)
where q(G) is half the dimension of the symmetric space attached to G.
23
Endoscopy – III.3
III.3 – Stable conjugacy and κ-orbital integrals
We shall now define stable conjugacy in general. Let G be a connected reductive group
over some field F . For simplicity we assume F is a field of characteristic zero.
We denote by GSC the simply connected cover of its derived subgroup. Given γ ∈ G(F )
we denote by Iγ the subgroup of G generated by Z(G) (the center of G) and the image in
G of the centralizer of γ in GSC . We call Iγ the stable centralizer of γ.
When γ is strongly regular semisimple i.e. if the centralizer is a torus, then Iγ is this
torus. If γ is regular semisimple i.e. if the centralizer has a torus as connected component,
then again Iγ is this torus. But in general Iγ is not a torus and can even be disconnected
as seen in the example of n0 ∈ SL(2, R) whose centralizer is ±N with N the group of
upper-triangular unipotent matrices. Nevertheless we have a
Lemma III.3 – Let γ be semisimple in G(F ). Then Iγ is a connected reductive subgroup,
namely the connected component of the centralizer.
Proof: The centralizer in GSC of a semisimple element γ is a connected reductive group;
this is due to Steinberg. Its image in G is again reductive and connected. It contains a
maximal torus T , but any T contains Z(G).
Consider γ and γ 0 in G(F ). We shall say that γ and γ 0 are stably conjugate if there is
an x ∈ G(F ) such that
(i)
x−1 γ x = γ 0
and
(ii)
aσ = xσ(x)−1 ∈ Iγ
for all σ ∈ Gal(F /F )
where Gal(F /F ) is the Galois group of F /F . In other words we assume that aσ is a
1-cochain with values in Iγ .
The same relations hold if we replace x by y = tx with t ∈ Iγ . The second condition is
automatically satisfied if the first is, whenever G = GSC : in fact aσ always belongs to the
centralizer of γ.
The pair (x, aσ ) defines a 0-cocycle with values in the quotient set
Iγ \G
and hence a class in the Galois cohomology set associated to this set
H 0 (Gal(F /F ), Iγ \G)
24
Endoscopy – III.3
also denoted
H 0 (F, Iγ \G)
or (Iγ \G)(F )
This is the set of rational points of the quotient Iγ \G. This cohomology set is the quotient
of the set of 0-hypercocycles as above by the equivalence relation:
(x, aσ ) ' (y, bσ )
whenever there exists t ∈ Iγ such that
y = tx and bσ = taσ σ(t)−1
We observe that there is an exact sequence of pointed sets
H 0 (F, Iγ \G) → H 1 (F, Iγ ) → H 1 (F, G)
induced by the map (x, aσ ) 7→ aσ and the inclusion Iγ → G. One denotes by D(F, Iγ \G)
the image of H 0 (F, Iγ \G) into H 1 (F, Iγ ) or equivalently the kernel of the next map
D(F, Iγ \G) = ker H 1 (F, Iγ ) → H 1 (F, G)
With this we have a small exact sequence of pointed sets
1 → Iγ (F )\G(F ) → (Iγ \G)(F ) → D(F, Iγ \G) → 1
To continue, let us assume first that γ is regular semisimple and hence Iγ is a maximal
torus T in G defined over F . We shall now introduce the abelian groups E(F, T \G). We
first recall that
D(F, T \G) = ker H 1 (F, T ) → H 1 (F, G)
the group E(F, T \G) is an abelianized version of it namely
1
E(F, T \G) = ker H 1 (F, T ) → Hab
(F, G)
There is a natural injective map
D(F, T \G) → E(F, T \G)
Observe that by composition we get a map
(T \G)(F ) → E(F, T \G)
This can also be seen as follows: we have
E(F, T \G) = Im H 1 (F, Tsc ) → H 1 (F, T )
25
Endoscopy – III.3
and that there is a natural map
(T \G)(F ) → H 1 (F, Tsc )
since H 1 (F, Tsc ) can be identified with an abelianized avatar of
H 0 (F, T \G) = (T \G)(F )
Assume from now on that F is a local field. When F is non archimedean the map
D(F, T \G) → E(F, T \G)
is bijective since
1
H 1 (F, G) → Hab
(F, G)
is bijective in this case. Over the reals, as we shall see in the next section, the set D(R, T \G)
is a subset but not usually a subgroup of E(R, T \G).
We shall denote by K(F, T \G) the Pontryagin dual of the finite abelian group E(F, T \G).
We call the elements of K(F, T \G) endoscopic characters. Let κ ∈ K(F, T \G) it defines
using the map
(T \G)(F ) → E(F, T \G)
a function, again denoted κ on (T \G)(F ). The function κ has the following multiplicative
property: assume that x and z = xy in G define elements in (T \G)(F ). This means that
xσ(x)−1 ∈ T and zσ(z)−1 ∈ T for all σ ∈ Gal(F /F ) then
yσ(y)−1 ∈ Tx := x−1 T x and κ(xy) = κx (y)κ(x)
where κx is the character of E(F, Tx \G) obtained from κ via the F -isomorphism
T → x−1 T x
A κ-orbital integral for γ regular in T (R) is defined by
Oγκ (f )
Z
κ(x)f (x−1 γ x) dẋ
=
(T \G)(F )
Implicit in the definition is a compatible choice of Haar measures on the various stable
conjugates x−1 T x of T for x ∈ (T \G)(F ) obtained by transporting invariant differential
forms of maximal degree.
The stable orbital integrals are the Oγ1 (f ) i.e. κ-orbital integrals when κ is trivial. They
are often denoted SOγ (f ).
26
Endoscopy – III.3
Now consider the case of an arbitrary semisimple element γ and let I = Iγ its stable
centralizer. We put again
D(F, I\G) = ker H 1 (F, I) → H 1 (F, G)
it has an abelianized avatar
1
1
E(F, I\G) = ker Hab
(F, I) → Hab
(F, G)
with a map
D(F, I\G) → E(F, I\G)
which need not be injective nor surjective in general. Now given a character κ of E(F, I\G)
the κ-orbital integral is defined as follows
Z
κ
Oγ (f ) =
e(Ix )κ(x)f (x−1 γ x) dẋ
(I\G)(F )
where e(Ix ) is the Kottwitz sign of the group
Ix := x−1 I x
the stable centralizer of x−1 γ x.
27
Endoscopy – III.4
III.4 – Stable conjugacy and compact Cartan subgroups over R
We shall now consider the special case where F = R and γ is elliptic and regular in G
which means that
T = Iγ
is an elliptic torus i.e. such that Tsc is an R-anisotropic torus.
Lemma III.4 – Let T be an elliptic torus in a real reductive group G. For any n in
NG (T ) the automorphism w = Ad(n) restricted to T is defined over R.
Proof: Any automorphism w of T is defined by a Z-linear automorphism w∗ of X ∗ (Tad );
it suffices to prove that w∗ is real. But σ the non trivial element in Gal(C/R) acts as -1
on X ∗ (Tad ) and hence commutes with w∗ .
We denote by ΩC (G, T ), or simply ΩG , the complex Weyl group, which is the group
generated by the automorphisms of T induced by Ad(n) with n ∈ NG (T ) and ΩR (G, T ),
or ΩK , the subgroup generated by the automorphisms of T induced by Ad(n) with
n ∈ G(R) ∩ NG (T ) = K ∩ NG (T )
where K is the maximal compact subgroup of G(R) containing T (R).
Proposition III.5 – Let T be an elliptic torus in G over R. There is a natural bijection
D(R, T \G) → ΩC (G, T )/ΩR (G, T )
Proof: Consider γ strongly regular in T and in particular T = Iγ . Then, we observe that
if γ 0 is stably conjugate to γ then, Iγ 0 is again an elliptic torus; but all elliptic tori are
conjugate in G(R). Hence, up to ordinary conjugacy we may replace γ 0 by γ 00 ∈ T and
now there is n in the normalizer NG (T ) of T in G such that
n−1 γ n = γ 00
The image of n in the Weyl group is uniquely defined by the pair (γ, γ 00 ) since γ is strongly
regular but the element γ 00 is defined by γ 0 only up to the action of the real Weyl group.
This yields an injective map
D(R, T \G) → ΩC (G, T )/ΩR (G, T )
Now the map is surjective thanks to lemma III.4.
28
Endoscopy – III.4
More generally we observe that all conjugacy classes inside the stable conjugacy class of
an elliptic element in T , intersect T .
Remark – The set D(R, T \G) is not usually a group. In fact ΩR (G, T ) is not an invariant
subgroup of ΩC (G, T ) in general although D(R, T \G) can be embedded naturally in the
abelian group E(R, T \G). For example consider G = U (2, 1), then
ΩC (G, T ) = S3
while
ΩR (G, T ) = S2
In this case D(R, T \G) has three elements and the group E(R, T \G) is isomorphic to
Z2 × Z2 .
29
Endoscopy – III.5
III.5 – Discrete series and stable conjugacy
We recall that, according to Harish-Chandra, the group G(R) has a discrete series representation if and only if there exist an R-elliptic torus T in G. Assume from now on this is
the case. They are parametrized as follows.
Given T , an R-elliptic torus T , we consider K a maximal compact subgroup of G(R)
containing T (R). Consider a Borel subgroup BK in KC containing T and a Borel subgroup
B in G containing BK . Now consider a character of T (R) defined by a
λ ∈ X ∗ (T ) ⊗ C
which is B-dominant. There is a discrete series representation πλ+ρ , where ρ is half the
sum of positive roots of T in B, characterized by its character given on T (R) by
P
w(λ+ρ)
w∈ΩK ε(w)e
q
P
Θλ+ρ = (−1)
wρ
w∈ΩG ε(w)e
where
1
( dim G(R) − dim K)
2
An L-packet of discrete series is a set of representations πµ where
q=
µ = w(λ + ρ)
with w ∈ ΩC
such that µ belongs to the Weyl chamber defined by BK . We shall denote by Ω(BK ) the
subset of such w. Clearly, Ω(BK ) is a set of representatives in ΩC (G, T ) of the quotient
ΩR (G, T )\ΩC (G, T ).
Proposition III.6 – The set of representations in an L-packet of discrete series is in
bijection with D(R, T \G); the bijection depends on the choice of B.
Proof: This follows from III.5 and the above remarks.
As already said the set D(R, T \G) embeds in the abelian group E(R, T \G). Thus we get
a pairing between the L-packet and K(R, T \G) the Pontryagin dual of E(R, T \G). This
pairing depends on the choice of B. We shall see that this pairing has another formulation
using L-groups.
Some choices of B seem to be better than others at least when G is a quasi-split group:
they are those that correspond to generic representations i.e. with a Whittaker model
(with respect to some further choice). For example, for G = U (2, 1) and K = U (2) × U (1)
once BK is chosen there are three Weyl chambers for G that are contained in the Weyl
chamber defined by BK in the Lie algebra of T . A Weyl chamber for G that contains ρK
is a “good” choice. There is only one for G = U (2, 1). But for G = SL(2), ρK = 0 and
any choice is good.
30
Endoscopy – III.6
III.6 – Pseudo-coefficients for discrete series
To simplify slightly the discussion assume that the center of G(R) is compact. Let πµ
be a discrete series representation of G(R) we say that a function f ∈ Cc∞ (G(R)) is a
(normalized) pseudo-coefficient for πµ if for any tempered irreducible representation π
we have
n
1 if π ' πµ
trace π(f ) =
0 otherwise
Observe that it would be more canonical to consider f dµ where dµ is a Haar measure in
G(R) since the choice of the Haar measure enters in the definition of trace π(f ).
The existence of pseudo-coefficients is an easy consequence of the existence of multipliers
due to Arthur [A] and Delorme [D]. We refer the reader to [CD] or [Lab2] for the construction. In this last paper more general functions, called index functions are constructed.
They allow in particular to deal with certain linear combinations of limits of discrete series
although individual ones do not have pseudo-coefficients.
A suitably normalized diagonal matrix coefficient of πµ (or rather its complex conjugate) would satisfy these requirements but for one condition: matrix coefficients are not
compactly supported unless the group is compact.
The interest of using pseudo-coefficients instead of matrix coefficients is that since they
are compactly supported trace π(f ) makes sense even if π is not tempered and they
can be used in the trace formula. These two properties are tied up since non tempered
representation show up in the spectral decomposition of the space of square integrable
automorphic forms.
We shall denote fµ a pseudo-coefficient for πµ , although it is highly non unique. But as
regards invariant harmonic analysis this plays no role. In particular the orbital integrals
are independent of the choice of the pseudo-coefficient; they are also independent of the
choice of the Haar measure on G(R) but one has to use the canonical measure on the
maximal compact torus.
The orbital integrals of fµ are easily computed for γ regular semisimple:
Oγ (fµ ) =
Θµ (γ −1 )
0
where Θµ is the character of πµ .
31
if γ is elliptic
otherwise
Endoscopy – III.7
III.7 – The dual picture
Let G be a reductive connected algebraic group and let G∗ be its quasi-split inner form.
ˇ with a Galois action. Then, Ǧ is the
To G∗ is associated a based root datum (X, ∆, X̌, ∆)
ˇ X, ∆).
complex reductive Lie group associated to the dual based root datum (X̌, ∆,
The Galois goup acts on the based root datum and hence on Ǧ via holomorphic automorphisms that will be moreover assumed to preserve a splitting. The L-group attached
to G is the semi-direct product
L
G = Ǧ o WR
here WR acts via its quotient Gal(C/R).
A Langlands parameter is an homomorphism
ϕ : WR → L G
whose image contains only semisimple elements and yields the identity when composed
with the projection on WR .
To each conjugacy class of Langlands parameter is associated an L-packet of representations of G(R). We have explained this correspondence for GL(2) and SL(2) above. The
general case has been established by Langlands in a paper unpublished for a long time and
now available in print [Lan2]. One denotes by Π(ϕ) the L-packet attached to ϕ.
Let Sϕ denote the centralizer in Ǧ of ϕ(WR ) and denote by Sϕ the quotient of Sϕ by
its connected component Sϕ0 times Z(Ǧ)Γ the center of L G. In [She4] Diana Shelstad has
established the
Proposition III.7 – For discrete series parameters there is an isomorphism between Sϕ
and K(R, T \G).
We give the proof when G is semisimple and simply connected. Consider an elliptic torus
T in G. Let Ť be the goup of complex characters of the lattice X̌(T ). Now consider a
discrete series parameter ϕ. We may and will assume that Ť contains ϕ(C× ) the image of
the subgoup C× ⊂ WR but then Sϕ is the subgroup of invariant elements in Ť under the
Galois action from L T and hence Sϕ is the set of elements of order 2 in Ť . We observe that
an element in K(R, T \G) defines a complex character of X̌(T ). Hence to κ ∈ K(R, T \G)
corresponds an element s ∈ Ť that moreover commutes with the Galois action. We thus
get a bijective homomorphism
K(R, T \G) → Sϕ
For p-adic fields the Langlands parametrization and the structure of L-packets of discrete
series are not known in general but it can be checked in examples that the group Sϕ is
often bigger than K and can even be non abelian.
32
Endoscopy – III.8
III.8 – Endoscopic groups
We have to recall the Tate-Nakayama isomophism for tori. Let S be a torus defined over
F and split over a Galois extension K; we denote by X̌(S) the Z-free module of finite rank
of its cocharacters.
Theorem III.8 – Assume that F is a local field. There is a canonical isomorphism
between Tate cohomology groups
Ĥ i (K/F, X̌(S)) → Ĥ i+2 (K/F, S(K))
In particular there is a canonical isomorphism
Ĥ −1 (K/F, X̌(S)) → H 1 (K/F, S(K)) = H 1 (F, S)
where, by definition
Ĥ −1 (Γ, X) := Ĥ0 (Γ, X)) = X NΓ /IΓ X
where X NΓ is the kernel of the norm endomorphism of X:
X
NΓ : x 7→
σ(x)
σ∈Γ
and IΓ is the augmentation ideal in the group algebra Z[Γ]
)
(
X
X
nσ = 0
IΓ = τ =
nσ σ σ∈Γ
σ∈Γ
Altogether we have the
Proposition III.9 – There is a canonical isomorphism
X̌(S)NΓ /IΓ X̌(S) → H 1 (F, S)
with Γ = Gal(K/F ).
Now there is a surjective map
H 1 (F, Tsc ) → E(F, T \G)
and if moreover we assume T elliptic then Tsc is anisotropic and hence we have a bijection
X̌(Tsc )NΓ → X̌(Tsc )
33
Endoscopy – III.8
The above proposition yields a surjective map
X̌(Tsc ) → E(F, T \G)
and hence it T is elliptic any character κ of E(F, T \G) defines a character κ of X̌(Tsc ).
Let RG and ŘG denote the set of roots and coroots of T in G. We observe that the
coroots are homomorphisms of the multiplicative group Gm into T that factor through Tsc
and hence
ŘG ⊂ X̌(Tsc )
Let us denote by Řκ the subset of coroots α̌ ∈ Ř such that
κ(α̌) = 1
Lemma III.10 – This is a (co)root system
Proof: In fact Řκ is the root system for the connected centralizer Ȟ of the image s of κ in
Ť ⊂ Ǧ.
Let X = X(T ) and X̌ = X̌(T ), there is a root datum
(X, Rκ , X̌, Řκ )
which inherits, from the Galois action on T , a natural Galois action and we get a quasisplit reductive group denoted Hκ (or simply H) with maximal torus TH ' T and coroot
system
ŘH ' Řκ
This is the elliptic endoscopic group attached to κ.
More generally, if we do not assume T elliptic, one can associate, as above, an endoscopic
group Hκ to any characters κ of X̌(Tsc )/IΓ X̌(Tsc ). Such κ’s naturally occur through localglobal constructions for the stabilization of the trace formula. Now κ defines, by restriction
to X̌(Tsc )NΓ , a character κ of
H 1 (F, Tsc )
which descends to a character of E(F, T \G) if κ is trivial on the kernel of the map
H 1 (F, Tsc ) → H 1 (F, T)
and in all cases κ defines a function on (T \G)(F ) via the natural map
(T \G)(F ) → H 1 (F, Tsc )
34
Endoscopy – III.8
It gives rise, when κ is non trivial on the above kernel, to the variant of ordinary endoscopy
that deals with orbital integrals weighted by a character ω as in II.3 above.
We observe that H and G share a torus and that the Weyl group Ω(TH , H) is canonically
isomorphic to a subgroup of Ω(T, G). Nevertheless there may not exist any homomorphism
from H to G that extends the isomorphism TH ' T .
Consider the image s of κ in Ť , its connected centralizer is Ȟ. An admissible embedding
of L H in L G is an L-homomorphism
η : LH → LG
that extends the natural inclusion
Ȟ → Ǧ
(by L-homomorphism we understand an homomorphism whose restriction to Ȟ is holomorphic and induces the identity on WR ).
A sufficient condition for the existence of an embedding is that the center of Ǧ is connected. There are examples where there is no admissible embedding.
35
Endoscopy – III.9
III.9 – Endoscopic transfer
Consider T an elliptic torus and κ an endoscopic character. Let H be the endoscopic group
defined by (T, κ). We need some notation. We let BG be a Borel subgroup of G containing
T.
Y
∆B (γ) =
(1 − γ −α )
α>0
where the product is over the positive roots for for the order defined by B. There is only
one choice of a Borel subgroup BH in H, containing TH compatible with the isomorphism
j : TH ' T .
Proposition III.11 – Assume there is an admissible embedding
η : LH → LG
One can attach to the triple (G, H, η) a character χG,H of T (R) with the following property.
Given f a pseudo-coefficient for a discrete series on G, there is a function f H which is a
linear combination of pseudo-coefficients for discrete series on H such that for a γ = j(γH )
regular in T (R)
κ
SOγH (f H ) = ∆G
H (γH , γ)Oγ (f )
where ∆G
H (γH , γ) the “transfer factor” has the following expression:
−1 −1
(−1)q(G)+q(H) χG,H (γ)∆B (γ −1 ) . ∆BH (γH
)
We shall give, in the next section, a proof of this proposition when G(R) = U (2, 1) and
in a further section in general. The expression for the transfer factor is borrowed from
Kottwitz [Ko2].
The character χG,H depends on the choice of the admissible embedding η. To obtain an
unconditional and more canonical transfer we may change a little the requirements: using
the “canonical” transfer factor one gets, instead of a transfer to H(R), a transfer to H1 (R)
some covering group of H(R) and thus one gets functions f H1 that transform according
to some character of the kernel of
H1 (R) → H(R)
If an admissible embedding exists the character on the kernel is the restriction of some
character of H1 (R) which allows to twist the transfer so that it descends to H(R). This
will be explained in the case G = U (2, 1) below.
The transfer f 7→ f H of pseudo-coefficients can be extended to all functions in Cc∞ (G(R));
to define it one has to extend the correspondence γ 7→ γH , called the norm, to all semisimple
36
Endoscopy – III.9
regular elements and the definition of the transfer factors to all tori. This is established
by Shelstad in the series of four papers [She1], [She2], [She3] and [She4].
Theorem III.12 – Assume there is an admissible embedding
η : LH → LG
∞
One can define transfer factors ∆G
H (γH , γ) such that for any f ∈ Cc (G(R) there exists a
function f H ∈ Cc∞ (H(R) with
κ
SOγH (f H ) = ∆G
H (γH , γ)Oγ (f )
whenever γH is a norm of γ semisimple regular and
SOγH (f H ) = 0
if γH is not a norm.
It should be observed that the correpondence
f 7→ f H
is not a map since f H is not uniquely defined. In fact f H is only prescribed through its
orbital integrals. But as regards invariant harmonic analysis this is a well defined object.
The geometric transfer
f 7→ f H
is dual of a transfer for representations. To any admissible irreducible representation σ of
H(R) it corresponds an element σG in the Grothendieck group of virtual representations of
G(R) as follows. Given ϕ a Langlands parameter for H then η ◦ϕ is a Langlands parameter
for G∗ where η is the admissible embedding
η : LH → LG
as above. Now consider Σ the L-packet of admissible irreducible representation of H(R)
corresponding to ϕ and Π the L-packet of representations of G(R), corresponding to η ◦ ϕ
(that can be the empty set if this parameter is not relevant for G).
Theorem III.13 – There is a function
ε : Π → ±1
such that, if we consider σG in the Grothendieck group defined by
X
σG =
ε(π) π
π∈Π
37
Endoscopy – III.9
then σ 7→ σG is the dual of the geometric transfer:
trace σG (f ) = trace σ(f H )
This is the Theorem 4.1.1 of [She4]. We shall prove it for discrete series of U (2, 1) below.
We shall now give an expression for ε(π) following section 5 of [She4].
Suppose we are given a complete set of inequivalent endoscopic groups H and for each
H an admissible embedding
η : LH → LG
Now consider a parameter
ϕ : WR → L G
The connected centralizer of s ∈ Sϕ is a group Ȟs conjugate to Ȟ for some H and hence
a conjugate of ϕ factors through η(L H) and defines an L-paquet Σs of representations of
H(R). (The L-packet is not unique in general: it depends on the choice of the conjugate
which may not be unique). On the other hand, using proposition III.7 above Shelstad
defines a pairing < s, π > between Sϕ and Π(ϕ) and shows that
ε(π) = c(s) < s, π >
and hence the above identity
X
trace σ(f H ) =
σ∈Σ
X
ε(π) trace π(f )
π∈Π
reads
e s (f H ) =
Σ
X
< s, π > trace π(f )
π∈Π
where
e s (f H ) = c(s)−1
Σ
X
trace σ(f H )
σ∈Σs
This can be inverted to give the theorem 5.2.9 of [She4]:
Theorem III.14 –
trace π(f ) =
1 X
e s (f H )
< s, π > Σ
#Sϕ
s∈Sϕ
38
Endoscopy
IV. A second example:
39
U (2, 1)
Endoscopy – IV.1
IV.1 – Discrete series and transfer for U (2, 1)
Consider the quasisplit unitary group in three variables G(R) = U (2, 1). A maximal
compact torus T (R) in G(R) is such that
T (R) ' U (1)3
and can be represented by matrices

u cos θ

0
g(u, v, w) =
iu sin θ
0
v
0

iu sin θ
0 
u cos θ
where u and v are complex numbers with |u| = |v| = 1 and w = r(θ). More precisely
(u, v, w) 7→ g(u, v, w)
defines a twofold cover T1 of T . The set D(R, T, G) is isomorphic to S3 /S2 and E(R, T, G)
is isomorphic to (Z2 )2 . The root system can be represented by the αi,j = φi − φj with
eiφ1 = ueiθ
eiφ2 = v
eiφ3 = ue−iθ
We take as positive roots those with i < j. We observe that the half sum of the roots
verifies ρ = α1,3 .
Choose as a maximal compact subgroup K in G(R)


λ+ν

1 √
2τ
K= k=

2
λ−ν
√
− 2σ
√2µ
2σ

λ√
−ν 
− 2τ 

λ+ν
where k is a conjugate of

λ
0
0
0
µ
σ

0
τ  ∈ U (1) × U (2) ⊂ U (3)
ν
It has been so chosen that it contains T (R). The Weyl group of K is generated by w0 the
symmetry with respect to ρ.
Now, consider a dominant weight λ. One attaches to it a discrete series representation
πµ with
µ=λ+ρ
40
Endoscopy – IV.1
The L-packet containing πµ contains also the various πwµ . Moreover πwµ is equivalent to
πw0 wµ and hence we may parametrize the L-packet by the elements in the Weyl group
such that sign (w) = 1. For such a w the character of πwµ is given by
Θwµ (γ) =
γ w(µ) − γ w0 w(µ)
γ ρ ∆B (γ)
with
∆B (γ) =
Y
(1 − γ −α )
α>0
Consider κ 6= 1 such that κ(α̌1,3 ) = 1. Such a κ is unique: in fact one has necessarily
κ(α̌1,2 ) = κ(α̌2,3 ) = −1
The endoscopic group H one associates to κ is isomorphic to
U (1, 1) × U (1)
the positive root of T in H (for a compatible order) being
α1,3 = ρ
The group H can be embedded in G as the subgroup of matrices in G of the form


ua
0 iub
g(u, v, w) =  0
v 0 
−iuc 0 ud
with
w=
a b
c d
and ad − bc = 1
It will be useful to consider also the twofold cover
H1 = U (1) × U (1) × SL(2)
(with maximal torus T1 ) defined by
(u, v, w) 7→ g(u, v, w)
Let fµ be a pseudo-coefficient for πµ then the κ-orbital integral of a γ regular in T (R) is
given by
X
−1
Oγκ (fµ ) =
κ(w)ΘG
µ (γw )
sign (w)=1
41
Endoscopy – IV.1
but using the bijection
ΩG /ΩK ' ΩK \ΩG
this can be rewritten
X
Oγκ (fµ ) =
κ(w)Θwµ (γ −1 )
sign (w)=1
The transfer factor ∆(γ, γH ) is given by
−1 −1
(−1)q(G)+q(H) χG,H (γ)∆B (γ −1 ) . ∆BH (γH
)
for some character χG,H defined as follows. It would be canonical to take
χG,H (γ −1 ) = γ ρ−ρH
but this is does not make sense since ρH defines a character of the twofold cover T1 (R) but
not of T (R). A way out is to consider a weight ξ that defines a character of the cocenter
of the twofold cover H1 (R) of H(R) so that
ρ−ρH +ξ
χ−1
G,H = e
descends to a character of T (R). With such a choice we get when sign (w) = 1
w(µ)+ξ
∆(γ
−1
−1
, γH
)ΘG
wµ (γ)
=−
γH
w w(µ)+ξ
− γH0
γ ρH ∆BH (γH )
We observe that κ(w) = −1 if sign (w) = 1 and w 6= 1 and that wµ is positive or negative
with respect to BH according to the sign of κ(w). Hence
−1
−1
∆(γ, γH )ΘG
) = κ(w)−1 SΘH
wµ (γ
ν (γH )
where SΘH
ν is the character of the L-packet Σν of discrete series for H defined by the
parameter
ν = w(µ) + ξ
Altogether we get
∆(γ, γH )Oγκ (fµ ) =
X
−1
SΘH
ν (γH )
ν
the sum being over the ν = w(µ) + ξ with sign (w) = 1 and this can be rewritten
X
∆(γ, γH )Oγκ (fµ ) =
SOγH (gν )
ν
where gν is a pseudo-coefficient for any one of the discrete series of H(R) in the L-packet
Σν attached to ν. In other words we have
X
fµH =
gν
ν
42
Endoscopy – IV.1
Notice that the set of parameters ν depends on the choice of ξ. Alternatively we could
use the canonical choice
χG,H (γ −1 ) = γ ρ−ρH
if we replace H by H1 and then the set of parameters would be canonical. This second solution is not the classical one but it seems after all more natural and at any rate considering
central extensions H1 instead of H cannot be avoided in the twisted case.
Now more generally we get
H
fwµ
=
X
a(w, ν)gν
ν
with
a(w1 , w2 µ) = κ(w2 )κ(w2 w1 )−1 .
Since pseudo-coefficients give a dual basis of the set of characters, in the space they generate, we get
X
trace Σν (f H ) =
a(w, ν) trace πwµ (f )
w
at least when f is a linear combination of pseudo-coefficients. Now, observe that
a(w1 , w2 µ) = κ(w2 )κ(w2 w1 )−1 = κw2 (w1 )−1
Let us denote by κν the character
x 7→ κw (x)−1
when ν = w(µ) + ξ. Now, using the bijection
Πµ ' D(R, T, G)
we get a pairing denoted < , > between Πµ and K(R, T, G) and the transfer equation can
be written
X
trace Σν (f H ) =
< κν , π > trace π(f )
π∈Πµ
The above formula for the dual transfer can be rewritten
X
trace Σs (f H ) =
< s, π > trace π(f )
π∈ΠΣ
where the pairing < s, π > is the Shelstad pairing between
Sϕ ' K(R, T, G)
and Πµ .
43
Endoscopy – IV.2
IV.2 – The dual picture for U (2, 1)
Let us now describe the dual picture. Observe that κ is the image, via Tate-Nakayama
duality, of


1 0 0
s =  0 −1 0  ∈ Ť ⊂ Ǧ
0 0 1
and hence


∗ 0 ∗
Ȟ =  0 ∗ 0 
∗ 0 ∗
The holomorphic Galois action on Ǧ is given by
g 7→ J t g −1 J −1
with

0 0
J =  0 −1
1 0

1
0
0
t −1 −1
h JH with
while on Ȟ it is h 7→ JH

JH
0

=
0
−1

0 1
1 0
0 0
As a consequence of the difference between J and JH there is no canonical admissible
embedding of L H in L G.
44
Endoscopy – IV.3
IV.3 – Discrete transfer
Consider a reductive Lie group G(R) with compact maximal torus T (R). In particular,
G(R) has discrete series. We choose an endoscopic character κ, this defines an endoscopic
group H. We choose a Borel subgroup B in G containing T and a compatible Borel
subgroup BH in H. To simplify the discussion we assume that ρ − ρH the difference of
half sums of positive roots for G and H respectively defines a character of T (R) and hence
the canonical transfer factor:
P
ε(w)γ wρ
−1
q(G)−q(H) P w∈ΩG
∆(γ ) = (−1)
wρH
w∈ΩH ε(w)γ
is a well defined function.
Let µ be parameter for a discrete series and let fµ be a pseudo-coefficient for πµ . Let
w0 ρ be the half sum of positive roots for the Weyl chamber defined by µ i.e. µ = w0 µ0
where µ0 is B-dominant and regular. We have
X
(−1)q(G) Oγκ (fˇµ ) = (−1)q(G)
κ(w)Oγw (fˇµ )
w∈ΩG /ΩK
X
=
w∈ΩG /ΩK
P
0 ww0 µ
w0 ∈ΩK ε(w )γ
κ(w) P
ww1 w0 ρ
w1 ∈ΩG ε(w1 )γ
so that
q(G)
(−1)
Oγκ (fˇµ )
P
= ε(w0 )
κ(w)ε(w)γ wµ
wρ
w∈ΩG ε(w)γ
w∈ΩG
P
Hence, we have
q(H)
(−1)
∆(γ
−1
)Oγκ (fˇµ )
P
wµ
w∈ΩG κ(w)ε(w)γ
= ε(w0 ) P
wρH
w∈ΩH ε(w)γ
Lemma IV.1 – The function κ is left invariant under ΩH :
Proof: Let w ∈ ΩG and w0 ∈ ΩH then the multiplicativity property of κ shows that
κ(w0 w) = κw0 (w)κ(w0 )
But, by definition of H, for any reflexion sα ∈ ΩH one has
κ(sα ) = κ(α̌) = 1
and κsα = κ
and hence, by induction on the length of w0 , one has κ(w0 ) = 1 and κw0 (w) = κ(w) so
that
κ(w0 w) = κ(w)
45
Endoscopy – IV.3
Now introduce Ω∗ (µ) the set of representatives of the quotient ΩH \ΩG such that w∗ µ is
BH dominant. The above lemma shows that
=
X
(−1)q(H) ∆(γ −1 )Oγκ (fˇµ )
P
ε(w)γ ww∗ µ
κ(w∗ )ε(w∗ w0 ) Pw∈ΩH
wρH
w∈ΩH ε(w)γ
w∗ ∈Ω∗ (µ)
Altogether, if we denote by gν a pseudo-coefficient for the discrete series of H(R) with
parameter ν we have
X
∆(γ )Oγκ (fµ ) =
κ(w∗ )ε(w∗ w0 )SOγ (gw∗ µ )
w∗ ∈Ω∗ (µ)
and hence
fµH =
X
a(w, µ) gwµ
w∗ ∈Ω∗ (µ)
with
a(w, µ) = κ(w∗ )ε(w∗ w0 )
and where w is any element in the ΩH -class defined by w∗ ∈ Ω∗ (µ).
Consider ν = w∗ µ = w∗ w0 µ0 and let Σ be the L-packet of representations of H(R) with
parameter ν. We have wΣ = w∗ w0 ∈ Ω∗ (µ0 ) and hence
a(wΣ , µ0 ) = κ(wΣ )ε(wΣ ) = κwΣ (w0−1 )−1 κ(w∗ )ε(wΣ )
so that
a(w, µ) = κwΣ (w0−1 ) a(wΣ , µ0 )
Let
< κΣ , π >= κwΣ (w0−1 )
where π is the discrete series for G with parameter µ = w0 µ0 ; then
a(w, µ) =< κΣ , π > c(Σ, π0 )
with
c(Σ, π0 ) = a(ww0 , µ0 )
and π0 defined by µ0 ; altogether we get
X
fπH =
< κΣ , π > c(Σ, π0 )gσ
Σ
the sum being over L-packets Σ of discrete series for H(R) with parameters in the orbit of
µ0 and we have chosen some σ ∈ Σ. This is equivalent to
X
c(Σ, π0 )−1 trace Σ(f H ) =
< κΣ , π > trace π(f )
π
the sum being over discrete series for G(R) with parameters in the orbit of µ0 .
46
Endoscopy
V. Further developments
47
Endoscopy – V.1
V.1 – K-groups
To get a more uniform treatment of endoscopy when F is any local field it is better to introe associated
duce, following Adams-Barbasch-Vogan Kottwitz and Arthur, the K-group G
to G. This is a disjoint union of groups indexed by
A = Im [H 1 (F, GSC ) → H 1 (F, G)]
The group Gα above α ∈ A is the inner form of G defined by α (or rather its image in the
adjoint group). There are also maps between Gβ and Gα :
ψαβ : Gβ → Gα
satisfying the obvious composition rules and such that ψαβ defines Gβ as an inner form of
Gα : we are given 1-cocycles uαβ;σ ∈ Gα with
ψαβ σ(ψαβ )−1 = Int uαβ;σ
and whose cohomology class belong to the image of the H 1 of the simply connected group
e = G but for F = R this is not so in general.
Gα,SC . When F is non archimedean one has G
e associated to G is the set of
A stable conjugacy class of γ ∈ Gα (F ) in the K-group G
γ ∈ Gβ (F ) for some β and such that there is x ∈ Gα with
0
ψαβ (γ 0 ) = x−1 γx and aσ = xuαβ;σ σ(x)−1 ∈ Iγ
The set of classes of such 1-cocycles a build a set we denote by
e
D(F, Iγ \G)
Assume that γ is regular semisimple, then Iγ is a torus T .
Proposition V.1 – Let F be a local field and let T be a torus. There is a bijective map
e → E(F, T \G)
D(F, T \G)
Proof: Recall that
1
E(F, T \G) = ker[H 1 (F, T ) → Hab
(F, G)]
By definition of the abelianized cohomology the class of a 1-cocycle a with values in T has
1
a trivial image in Hab
(F, G) if and only if
aσ = xuσ σ(x)−1 ∈ T
for some x ∈ G and some 1-cocycle u image of a 1-cocycle with values in GSC . Hence the
map is well defined and is obviously bijective.
48
Endoscopy – V.2
V.2 – The twisted case
Another genralization is important for applications: the twisted case. We consider a group
G acting on the left on a space L
(x, δ) 7→ x δ
, x∈G
δ∈L
which is a principal homogeneous space under this action. We say that L is a twisted
G-space if we are given a G-equivariant map
AdL : L → Aut G
This allows to define a right action of G on L such that
x δ y = x (AdL (δ) y) δ
By choosing a point δ0 ∈ L we get an isomorphism
L→Goθ
where
θ = AdL (δ0 )
but this isomorphism is not canonical when G has a non trivial center. This is why it is
better to work with L rather than with G o θ. One can develop as above the notion of
stable conjugacy inside L(F ) once stable centralizers are defined: given δ ∈ L(F ) we say
that δ 0 ∈ L(F ) is stably conjugate if there is x ∈ G(F ) such that
x−1 δx = δ 0
and xσ(x)−1 ∈ Iδ
for all σ ∈ Gal(F /F ) and where Iδ is the group generated by ZL := Z(G)θ and the image
of GδSC the fixed points in GSC under the adjoint action of δ. The stable centralizer Iδ
may not be connected even if δ is regular semisimple i.e. when GδSC is a torus.
We refer to [KS], [Lab3] and [Lab4] for a study of the twisted case in the trace formula
context. As regards the real case Shelstad’s results have been extended to the twisted case
by Renard [Ren].
Remark – We warn the reader that the definition given for the stable centralizer in [Lab3]
differs slightly from the one given above and used in [Lab4].
49
Endoscopy – V.3
V.3 – Trace formula stabilization
Last but not least the main motivation for studying endoscopy is the adelic trace formula.
In the old paper [LL] the trace formula for SL(2) was expressed as a sum of stable trace
formulas for its endoscopic groups. The stabilization gives a better understanding of the
space of automorphic forms and allows to establish cases of Langlands functoriality. This
was the beginning of all this game.
Later on various other instances of endoscopy and of twisted endoscopy were treated like
the Base Change for GL(2) by Saito-Shintani and Langlands [Lan1] and for unitary groups
in three variables by Rogawski [Rog]. Endoscopy also played a role at the place where it
was first discovered: for computing the Zeta function of Shimura varieties.
At the present time the problem of stabilizing the trace formula which is to express the
trace formula for G (or more generally the trace formula for a twisted space) as a linear
combination of stable trace formulas for its endoscopic groups is not completely solved.
Wonderful progress have been made by Jim Arthur in first establishing the trace formula
for arbitrary reductive groups and then by developping techniques that have allowed him
to get a conditional stabilization of the (non twisted) trace formula. The condition is about
the endoscopic transfer for non archimedean fields. Observe that transfer factors have been
defined for all local fields by Langland and Shelstad [LS]. One needs the existence of the
transfer: given f there exists f H satisfying the transfer identity recalled below and one
needs also that the so-called fundamental lemma holds.
The fundamental lemma: Assume that f is the characteristic function of an hyperspecial maximal compact subgroup in G(F ) (when this makes sense) and let f H be the
similar function for H. Then the transfer identity holds:
κ
SOγH (f H ) = ∆G
H (γH , γ)Oγ (f )
whenever γH is a norm of γ and it vanishes if γH is not a norm.
In fact one also needs a weighted variant of it. These questions have resisted the attacks
until recently although many important advances have been made by many people among
which I should mention Waldspurger in particular: he has proved that the non archimedean
transfer holds if the fundamental lemma is true.
A proof of the fundamental lemma for unitary groups has been obtained last year by
Laumon and Ngô Bao Châu. But the weighted version of it is still lacking to finish the
stabilization even for unitary groups and of course one needs the twisted analogues if one
wants to deal with the twisted case.
Things are now progressing faster. For example recent progress have been made by
Waldspurger to reduce the twisted case to the ordinary one.
50
Endoscopy – V.3
As a last word I would like to say that endoscopy is not the final word and some are
looking
beyond endoscopy
but this is another story.
51
Endoscopy
VI. Bibliography
52
Endoscopy
References
[Bo] M. V. Borovoi, Abelian Galois cohomology of Reductive groups, Mem. Amer. Math.
Soc. 132 (1998), No. 626.
[A] J. Arthur, A Paley-Wiener theorem for real reductive groups, Acta Math. 150 (1983),
1-89.
[CD] L. Clozel P. Delorme, Pseudo-coefficients et cohomologie des groupes réductifs, C.R.
Acad. Sci. Paris 300 No. 11 (1985), 385-387.
[D] P. Delorme, Multipliers for the convolution algebra of left and right K-finite compactly
supported smooth functions on a semi-simple Lie group, Invent. Math. 75 (1984), 9-23.
[Ko1] R. Kottwitz, Sign changes in harmonic analysis on reductive groups, Trans. Amer.
Math. Soc. 278 (1983), p. 289-297.
[Ko2] R. Kottwitz, Shimura Varieties and λ-adic Representations, Perspectives in Math. 10
Academic Press (1988), pp. 161-210.
[KS] R. Kottwitz D. Shelstad, Foundations of Twisted Endoscopy, Astérisque 255 (1999).
[Lab1] J.-P. Labesse, Cohomologie, L-groupes et fonctorialité, Compositio 55 (1985), p. 163184.
[Lab2] J.-P. Labesse, Pseudo-coefficients très cuspidaux et K-théorie, Math. Ann. 291 (1991),
p. 607-616.
[Lab3] J.-P. Labesse, Cohomologie, Stabilisation et changement de base , Astérisque 257
(1999).
[Lab4] J.-P. Labesse, Stable twisted trace formula: elliptic terms, Journ. Inst. Math. Jussieu
3 (2004), p. 473-530.
[LL] J.-P. Labesse R.P. Langlands, L-indistinguishability for SL2 , Can. J. Math. 31
(1979), p. 726-785.
[Lan1] R.P. Langlands, Base Change for GL(2), Annals of Math. Studies 96 Princeton Univ.
Press (1980).
[Lan2] R.P. Langlands, On the classification of irreducible representations of real algebraic
groups, Math. Surveys and Monographs 31 AMS (1989). p 101-170
[LS] R.P. Langlands D. Shelstad, On the Definition of Transfer Factors, Math. Annalen
278 (1987). p 219-271
[Ren] D. Renard, Twisted endoscopy for real groups, Journ. Inst. Math. Jussieu 2 (2003),
p. 529-566.
[Rog] J. Rogawski, Automorphic Representations of Unitary Groups in Three Variables, Annals of Math. Studies 123 (1990).
[She1] D. Shelstad, Characters and inner forms of a quasi-split group over R, Compositio
Math. 39 (1979), p. 11-45.
[She2] D. Shelstad, Orbital integrals and a family of groups attached to a real reductive group,
Ann. Sci. E.N.S. 12 (1979), p. 1-31.
53
Endoscopy
[She3] D. Shelstad, Embeddings of L-groups, Can. J. Math. 33 (1981), p. 513-558.
[She4] D. Shelstad, L-Indistinguishability for Real Groups, Math. Annalen 259 (1982), p. 385430.
54